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Let $k$ be a field containing an algebraically closed field of characteristic
zero. If $G$ is a finite group and $D$ is a division algebra over $k$, finite
dimensional over its center, we can associate to a faithful $G$-grading on $D$
a normal abelian subgroup $H$, a positive integer $d$ and an element of
$Hom(M(H), k^\times)^G$, where $M(H)$ is the Schur multiplier of $H$. Our main
theorem is the converse: Given an extension $1\rightarrow H\rightarrow
G\rightarrow G/H\rightarrow 1$, where $H$ is abelian, a positive integer $d$,
and an element of $Hom(M(H), k^\times)^G$, there is a division algebra with
center containing $k$ that realizes these data. We apply this result to
classify the $G$-simple algebras over an algebraically closed field of
characteristic zero that admit a division algebra form over a field containing
an algebraically closed field.查看全文